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Maass Forms
In mathematics, Maass forms or Maass wave forms are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup \Gamma of \mathrm_(\R) as modular forms. They are Eigenforms of the hyperbolic Laplace Operator \Delta defined on \mathbb and satisfy certain growth conditions at the cusps of a fundamental domain of \Gamma. In contrast to the modular forms the Maass forms need not be holomorphic. They were studied first by Hans Maass in 1949. General remarks The group : G := \mathrm_(\R) = \left\ operates on the upper half plane :\mathbb = \ by fractional linear transformations: :\begin a & b \\ c & d \\ \end \cdot z := \frac. It can be extended to an operation on \mathbb \cup \ \cup \mathbb by defining: :\begin a & b \\ c & d \\ \end\cdot z :=\begin \frac & \text cz+d \neq 0, \\ \infty & \text cz+d=0,\end :\begin a & b \\ c & d \\ \end \cdot \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphic Form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups. Modular forms are holomorphic automorphic forms defined over the groups SL(2, R) or PSL(2, R) with the discrete subgroup being the modular group, or one of its congruence subgroups; in this sense the theory of automorphic forms is an extension of the theory of modular forms. More generally, one can use the adelic approach as a way of dealing with the whole family of congruence subgroups at once. From this point of view, an automorphic form over the group ''G''(A''F''), for an algebraic group ''G'' and an algebraic number field ''F'', is a complex-valued function on ''G''(A''F'') that is left ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Maass Form
In mathematics, a weak Maass form is a smooth function f on the upper half plane, transforming like a modular form under the action of the modular group, being an eigenfunction of the corresponding hyperbolic Laplace operator, and having at most linear exponential growth at the cusps. If the eigenvalue of f under the Laplacian is zero, then f is called a harmonic weak Maass form, or briefly a harmonic Maass form. A weak Maass form which has actually moderate growth at the cusps is a classical Maass wave form. The Fourier expansions of harmonic Maass forms often encode interesting combinatorial, arithmetic, or geometric generating functions. Regularized theta lifts of harmonic Maass forms can be used to construct Arakelov Green functions for special divisors on orthogonal Shimura varieties. Definition A complex-valued smooth function f on the upper half-plane is called a weak Maass form of integral weight (for the group ) if it satisfies the following three conditions: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henryk Iwaniec
Henryk Iwaniec (born October 9, 1947) is a Polish-American mathematician, and since 1987 a professor at Rutgers University. Background and education Iwaniec studied at the University of Warsaw, where he got his PhD in 1972 under Andrzej Schinzel. He then held positions at the Institute of Mathematics of the Polish Academy of Sciences until 1983 when he left Poland. He held visiting positions at the Institute for Advanced Study, University of Michigan, and University of Colorado Boulder before being appointed Professor of Mathematics at Rutgers University. He is a citizen of both Poland and the United States. He and mathematician Tadeusz Iwaniec are twin brothers. Work Iwaniec studies both sieve methods and deep complex-analytic techniques, with an emphasis on the theory of automorphic forms and harmonic analysis. In 1997, Iwaniec and John Friedlander proved that there are infinitely many prime numbers of the form . Results of this strength had previously been seen as co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors are Camillo De Lellis (Institute for Advanced Study, Princeton) and Jean-Benoît Bost (University of Paris-Sud Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anton Deitmar , the championship trophy of the Swedish junior hockey ...
Anton may refer to: People *Anton (given name), including a list of people with the given name *Anton (surname) Places *Anton Municipality, Bulgaria **Anton, Sofia Province, a village *Antón District, Panama **Antón, a town and capital of the district *Anton, Colorado, an unincorporated town *Anton, Texas, a city *Anton, Wisconsin, an unincorporated community *River Anton, Hampshire, United Kingdom Other uses *Case Anton, codename for the German and Italian occupation of Vichy France in 1942 *Anton (computer), a highly parallel supercomputer for molecular dynamics simulations * ''Anton'' (1973 film), a Norwegian film * ''Anton'' (2008 film), an Irish film *Anton Cup The Anton Cup is the championship trophy of the Swedish junior hockey league, J20 SuperElit. The trophy was donated by Anton Johansson, chairman of the Swedish Ice Hockey Association between 1924 and 1948, in 1952, as an award for Sweden's top-ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crelle's Journal
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by August Leopold Crelle (Berlin) in 1826 and edited by him until his death in 1855. It was one of the first major mathematical journals that was not a proceedings of an academy. It has published many notable papers, including works of Niels Henrik Abel, Georg Cantor, Gotthold Eisenstein, Carl Friedrich Gauss and Otto Hesse. It was edited by Carl Wilhelm Borchardt from 1856 to 1880, during which time it was known as ''Borchardt's Journal''. The current editor-in-chief is Rainer Weissauer (Ruprecht-Karls-Universität Heidelberg) Past editors * 1826–1856 August Leopold Crelle * 1856–1880 Carl Wilhelm Borchardt * 1881–1888 Leopold Kronecker, Karl Weierstrass * 1889–1892 Leopold Kronecker * 1892–1902 Lazarus Fuchs * 1903–1928 Kurt Hens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Voronoi Formula
In mathematics, a Voronoi formula is an equality involving Fourier coefficients of automorphic forms, with the coefficients twisted by additive characters on either side. It can be regarded as a Poisson summation formula for non-abelian groups. The Voronoi (summation) formula for GL(2) has long been a standard tool for studying analytic properties of automorphic forms and their ''L''-functions. There have been numerous results coming out the Voronoi formula on GL(2). The concept is named after Georgy Voronoy. Classical application To Voronoy and his contemporaries, the formula appeared tailor-made to evaluate certain finite sums. That seemed significant because several important questions in number theory involve finite sums of arithmetic quantities. In this connection, let us mention two classical examples, Dirichlet’s divisor problem and the Gauss’ circle problem. The former estimates the size of ''d''(''n''), the number of positive divisors of an integer ''n''. Di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Analytic Eisenstein Series
In mathematics, the simplest real analytic Eisenstein series is a special function of two variables. It is used in the representation theory of SL(2,R) and in analytic number theory. It is closely related to the Epstein zeta function. There are many generalizations associated to more complicated groups. Definition The Eisenstein series ''E''(''z'', ''s'') for ''z'' = ''x'' + ''iy'' in the upper half-plane is defined by :E(z,s) =\sum_ for Re(''s'') > 1, and by analytic continuation for other values of the complex number ''s''. The sum is over all pairs of coprime integers. Warning: there are several other slightly different definitions. Some authors omit the factor of ½, and some sum over all pairs of integers that are not both zero; which changes the function by a factor of ζ(2''s''). Properties As a function on ''z'' Viewed as a function of ''z'', ''E''(''z'',''s'') is a real-analytic eigenfunction of the Laplace operator on H with the eigenvalue ''s''(''s''-1). In other wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mock Modular Form
In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight . The first examples of mock theta functions were described by Srinivasa Ramanujan in his last 1920 letter to G. H. Hardy and in his lost notebook. Sander Zwegers discovered that adding certain non-holomorphic functions to them turns them into harmonic weak Maass forms. History Ramanujan's 12 January 1920 letter to Hardy listed 17 examples of functions that he called mock theta functions, and his lost notebook contained several more examples. (Ramanujan used the term "theta function" for what today would be called a modular form.) Ramanujan pointed out that they have an asymptotic expansion at the cusps, similar to that of modular forms of weight , possibly with poles at cusps, but cannot be expressed in terms of "ordinary" theta functions. He called functions with similar properties "mock theta functions". Zwegers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adele Ring
Adele Laurie Blue Adkins (, ; born 5 May 1988), professionally known by the mononym Adele, is an English singer and songwriter. After graduating in arts from the BRIT School in 2006, Adele signed a record deal with XL Recordings. Her debut album, '' 19'', was released in 2008 and spawned the UK top-five singles "Chasing Pavements" and "Make You Feel My Love". The album was certified 8× platinum in the UK and triple platinum in the US. Adele was honoured with the Brit Award for Rising Star as well as the Grammy Award for Best New Artist. Adele released her second studio album, '' 21'', in 2011. It became the world's best-selling album of the 21st century, with sales of over 31 million copies. It was certified 18× platinum in the UK (the highest by a solo artist of all time) and Diamond in the US. According to ''Billboard'', ''21'' is the top-performing album in the US chart history, topping the ''Billboard'' 200 for 24 weeks (the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
